Ponder This

June 2007

You are playing a game in which you can ask for numbers. Each such number is a random value uniformly (and independently) distributed between 0 and 1. After you receive each number you can decide whether to request an additional number or to stop. When you stop your score is the sum of all the numbers you have received. Let 0<x<1. We ask two questions.

Suppose you are trying to achieve a score between x and 1. What are your chances of success?

Suppose you are trying to achieve a score between n+x and n+1. What are your chances of success in the limit as n --> infinity?

In both cases assume you are following the obvious best strategy. The answers are simple functions of x. Both questions must be answered correctly for credit.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to:ponder@il.ibm.com